Upper bound for the number of zeros of a meromorphic function outside a vertical line and applications. (Majoration du nombre de zéros d’une fonction méromorphe en dehors d’une droite verticale et applications.) (French, English) Zbl 1203.11059

Velásquez Castañón, Oswaldo

J. Anal. Math. 110, 67-127 (2010).

Let \(a\in{\mathbb R}\), \(h(s)\) a meromorphic function in \({\mathbb C}\), real on \({\mathbb R}\), with only finitely many poles, finitely many zeros in the half-space \(\sigma>a\), holomorphic and non-zero on the critical line \(\sigma=a\). Let \(f(s)=f^{\pm}(s)=h(s)\pm h(2a-s)\). Suppose that the function \(F(s)=\frac{h(2a-s)}{h(s)}\) satisfies
(i) for every \(\eta>0\) there exists \(\sigma_0=\sigma_0(\eta)>a\) such that \(|F(s)|<\eta\) if \(\sigma\geq \sigma_0\) (\(s=\sigma+i\tau\));
(ii) for every \(\varepsilon>0\) and \(\sigma_0>a\) there exists a sequence \((T_n)_n \) such that \(\lim_{n\to\infty}T_n=\infty \) and \(|F(s)|<e^{\varepsilon |s|}\) for \(a\leq \sigma\leq \sigma_0,|\tau|=T_n,n\geq 1\).
The author proves that all zeros of \(f(s)\) up to a finite number lie on the line \(\sigma=a\) and are simple. He studies translations of the Riemann zeta-function and \(L\)-functions, and integrals of Eisenstein series, among others.

Reviewer: Florin Nicolae (Berlin)

MSC:

11M26

Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses

Keywords:

zeros of a meromorphic function outside a vertical line; Riemann zeta function; \(L\)-functions; Eisenstein series

Software:

PARI/GP

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